Question

the circle below has center o, and its radius is 4 cm. given that m∠aob=50°, find the length of the major arc acb. give an exact answer in terms of π, and be sure to include the correct unit in your answer. length of major arc acb :

Explanation:

Step1: Calcular el ángulo del arco mayor

El ángulo central total en un círculo es \( 360^\circ \). El ángulo del arco menor \( \angle AOB = 50^\circ \), así que el ángulo del arco mayor \( \angle AOCB \) es \( 360^\circ - 50^\circ = 310^\circ \).

Step2: Usar la fórmula de longitud de arco

La fórmula para la longitud de un arco es \( L = \frac{\theta}{360^\circ} \times 2\pi r \), donde \( \theta \) es el ángulo central y \( r \) es el radio. Aquí, \( \theta = 310^\circ \) y \( r = 4 \, \text{cm} \). Sustituyendo: \( L = \frac{310^\circ}{360^\circ} \times 2\pi \times 4 \). Simplificar: \( \frac{310}{360} \times 8\pi = \frac{310 \times 8\pi}{360} = \frac{62\pi}{9} \, \text{cm} \)? Espera, no, espera: \( \frac{310}{360} \times 2\pi \times 4 = \frac{310 \times 8\pi}{360} = \frac{62\pi}{9} \)? Wait, no, \( 2\times4=8 \), \( 310/360 = 31/36 \), entonces \( 31/36 \times 8\pi = (31\times8\pi)/36 = (248\pi)/36 = 62\pi/9 \)? Wait, no, 310-50=310? Wait, 360-50=310, sí. Entonces \( \theta = 310^\circ \), \( r = 4 \). Entonces \( L = \frac{310}{360} \times 2\pi \times 4 \). Simplificar \( 2\times4=8 \), \( 310/360 = 31/36 \), entonces \( 31/36 \times 8\pi = (31\times8\pi)/36 = 248\pi/36 = 62\pi/9 \)? Wait, no, 310 divided by 360 is 31/36, times 8π: 318=248, 248/36=62/9. So \( \frac{62\pi}{9} \, \text{cm} \)? Wait, no, wait, 31024=2480, 2480/360=62/9. Yes. So the length is \( \frac{62\pi}{9} \) cm? Wait, no, wait, 360-50=310, correct. Then arc length formula: \( L = \frac{\theta}{360} \times 2\pi r \). So \( \theta = 310 \), \( r = 4 \). So \( L = \frac{310}{360} \times 2\pi \times 4 = \frac{310 \times 8\pi}{360} = \frac{2480\pi}{360} = \frac{62\pi}{9} \) cm. Wait, but let's check again: 310/360 = 31/36, 2πr = 8π, so 31/36 8π = (318π)/36 = 248π/36 = 62π/9 ≈ 6.89π? Wait, no, 62 divided by 9 is about 6.89, but let's check the calculation again. Wait, 360-50=310, correct. Radius 4, so circumference is 8π. The major arc is 310/360 of the circumference. So 310/360 8π = (310*8π)/360 = 2480π/360 = 62π/9 cm. Yes.

Answer:

\( \frac{62\pi}{9} \, \text{cm} \)